9.1 Supersonic flow

Supersonic flow occurs when air moves faster than the speed of sound, creating unique phenomena like shock waves and expansion waves. These effects dramatically alter how air behaves around objects, impacting aircraft design and performance in ways that don't apply to slower speeds.

Understanding supersonic flow is crucial for designing high-speed aircraft and propulsion systems. Key concepts include the Mach number, shock wave formation, and how flow properties like pressure and temperature change abruptly across shock waves. This knowledge enables engineers to optimize supersonic vehicle designs.

Supersonic flow characteristics

• Supersonic flow occurs when the flow velocity exceeds the speed of sound, resulting in unique flow phenomena and compressibility effects
• The Mach number, defined as the ratio of flow speed to the local speed of sound, is a critical parameter in characterizing supersonic flow

Mach number vs flow speed

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• The Mach number ($M$) is calculated as $M = \frac{V}{a}$, where $V$ is the flow speed and $a$ is the local speed of sound
• Subsonic flow: $M < 1$, transonic flow: $M \approx 1$, supersonic flow: $M > 1$, hypersonic flow: $M >> 1$
• As the Mach number increases, compressibility effects become more significant, leading to changes in density, pressure, and temperature

Shock waves in supersonic flow

• Shock waves are thin regions of abrupt changes in flow properties that occur when the flow transitions from supersonic to subsonic speeds
• Shock waves can be normal (perpendicular to the flow) or oblique (inclined at an angle to the flow)
• Across a shock wave, pressure, density, and temperature increase, while velocity decreases

Expansion waves in supersonic flow

• Expansion waves occur when the flow expands and accelerates from a high-pressure region to a low-pressure region
• Expansion waves are isentropic and result in a decrease in pressure, density, and temperature, while increasing the Mach number
• Prandtl-Meyer expansion waves are a type of expansion wave that occurs at sharp corners or expansions in supersonic flow

Compressibility effects on density

• In supersonic flow, density changes significantly due to compressibility effects
• The density ratio across a shock wave is given by $\frac{\rho_2}{\rho_1} = \frac{(\gamma+1)M_1^2}{(\gamma-1)M_1^2+2}$, where $\rho$ is the density, $M_1$ is the upstream Mach number, and $\gamma$ is the specific heat ratio
• Density increases across shock waves and decreases across expansion waves

Temperature changes across shocks

• Temperature increases significantly across shock waves due to the compression and deceleration of the flow
• The temperature ratio across a normal shock wave is given by $\frac{T_2}{T_1} = \frac{[2\gamma M_1^2-(\gamma-1)][(\gamma-1)M_1^2+2]}{(\gamma+1)^2M_1^2}$, where $T$ is the temperature
• The temperature increase across shock waves is an important consideration in supersonic vehicle design, as it affects material selection and cooling requirements

Governing equations of supersonic flow

• The governing equations of supersonic flow are derived from the conservation laws of mass, momentum, and energy, along with the equation of state for the fluid
• These equations describe the behavior of compressible fluids and are essential for analyzing and predicting supersonic flow phenomena

Continuity equation for compressible flow

• The continuity equation represents the conservation of mass in a compressible flow
• For steady, one-dimensional flow, the continuity equation is given by $\rho_1 A_1 V_1 = \rho_2 A_2 V_2$, where $\rho$ is the density, $A$ is the cross-sectional area, and $V$ is the velocity
• The continuity equation relates changes in density, velocity, and area in a compressible flow

Momentum equation for compressible flow

• The momentum equation represents the conservation of momentum in a compressible flow
• For steady, one-dimensional flow, the momentum equation is given by $p_1 + \rho_1 V_1^2 = p_2 + \rho_2 V_2^2$, where $p$ is the pressure
• The momentum equation relates changes in pressure, density, and velocity in a compressible flow

Energy equation for compressible flow

• The energy equation represents the conservation of energy in a compressible flow
• For steady, one-dimensional flow with no heat addition or work extraction, the energy equation is given by $h_1 + \frac{V_1^2}{2} = h_2 + \frac{V_2^2}{2}$, where $h$ is the specific enthalpy
• The energy equation relates changes in enthalpy and kinetic energy in a compressible flow

Equation of state for ideal gases

• The equation of state relates the pressure, density, and temperature of a gas
• For an ideal gas, the equation of state is given by $p = \rho R T$, where $R$ is the specific gas constant
• The equation of state is used in conjunction with the governing equations to solve compressible flow problems

Normal shock waves

• Normal shock waves are shock waves that are perpendicular to the flow direction
• They occur when the flow encounters an obstruction or a sudden change in flow conditions, causing a rapid deceleration and compression of the flow

Normal shock wave properties

• Across a normal shock wave, the flow experiences an abrupt increase in pressure, density, and temperature, while the velocity decreases
• The flow downstream of a normal shock wave is always subsonic, regardless of the upstream Mach number
• The entropy of the flow increases across a normal shock wave, indicating that the process is irreversible

Rankine-Hugoniot equations

• The Rankine-Hugoniot equations are a set of relations that describe the changes in flow properties across a normal shock wave
• They are derived from the conservation of mass, momentum, and energy, along with the equation of state
• The Rankine-Hugoniot equations relate the upstream and downstream Mach numbers, pressures, densities, and temperatures

Mach number relations across normal shocks

• The upstream and downstream Mach numbers across a normal shock wave are related by $M_2^2 = \frac{1+\frac{\gamma-1}{2}M_1^2}{\gamma M_1^2-\frac{\gamma-1}{2}}$
• For a given upstream Mach number, the downstream Mach number can be calculated using this relation
• As the upstream Mach number increases, the downstream Mach number approaches a limiting value of $\sqrt{\frac{\gamma-1}{\gamma+1}}$

Pressure ratio vs Mach number

• The pressure ratio across a normal shock wave is a function of the upstream Mach number
• The pressure ratio is given by $\frac{p_2}{p_1} = 1+\frac{2\gamma}{\gamma+1}(M_1^2-1)$
• As the upstream Mach number increases, the pressure ratio across the shock wave also increases

Temperature ratio vs Mach number

• The temperature ratio across a normal shock wave is a function of the upstream Mach number
• The temperature ratio is given by $\frac{T_2}{T_1} = \frac{[2\gamma M_1^2-(\gamma-1)][(\gamma-1)M_1^2+2]}{(\gamma+1)^2M_1^2}$
• As the upstream Mach number increases, the temperature ratio across the shock wave also increases

Density ratio vs Mach number

• The density ratio across a normal shock wave is a function of the upstream Mach number
• The density ratio is given by $\frac{\rho_2}{\rho_1} = \frac{(\gamma+1)M_1^2}{(\gamma-1)M_1^2+2}$
• As the upstream Mach number increases, the density ratio across the shock wave also increases

Oblique shock waves

• Oblique shock waves are shock waves that are inclined at an angle to the flow direction
• They occur when the flow encounters a sharp corner or a wedge-shaped obstacle, causing a sudden deflection and compression of the flow

Oblique shock wave geometry

• The geometry of an oblique shock wave is characterized by the shock angle ($\beta$) and the deflection angle ($\theta$)
• The shock angle is the angle between the shock wave and the upstream flow direction
• The deflection angle is the angle through which the flow is turned by the shock wave

Oblique shock wave properties

• Across an oblique shock wave, the flow experiences an increase in pressure, density, and temperature, while the velocity decreases and the flow direction changes
• The flow downstream of an oblique shock wave can be either subsonic or supersonic, depending on the upstream Mach number and the shock angle
• The entropy of the flow increases across an oblique shock wave, indicating that the process is irreversible

Oblique shock wave equations

• The oblique shock wave equations relate the upstream and downstream flow properties, the shock angle, and the deflection angle
• The equations are derived from the conservation of mass, momentum, and energy, along with the equation of state
• The oblique shock wave equations can be used to calculate the downstream Mach number, pressure, density, and temperature for a given upstream Mach number and shock angle

Deflection angle vs shock angle

• For a given upstream Mach number, there is a unique relationship between the deflection angle and the shock angle
• This relationship is described by the $\theta$-$\beta$-$M$ relation, which is derived from the oblique shock wave equations
• The maximum deflection angle that can be achieved for a given upstream Mach number is called the shock detachment angle

Mach number relations across oblique shocks

• The upstream and downstream Mach numbers across an oblique shock wave are related by the oblique shock wave equations
• The downstream Mach number depends on the upstream Mach number, the shock angle, and the specific heat ratio of the gas
• As the shock angle increases, the downstream Mach number decreases, and the shock wave becomes stronger

Pressure ratio across oblique shocks

• The pressure ratio across an oblique shock wave is a function of the upstream Mach number and the shock angle
• The pressure ratio increases as the shock angle increases, indicating a stronger shock wave
• The pressure ratio across an oblique shock wave is always greater than 1, as the pressure increases across the shock

Density ratio across oblique shocks

• The density ratio across an oblique shock wave is a function of the upstream Mach number and the shock angle
• The density ratio increases as the shock angle increases, indicating a stronger shock wave
• The density ratio across an oblique shock wave is always greater than 1, as the density increases across the shock

Prandtl-Meyer expansion waves

• Prandtl-Meyer expansion waves occur when a supersonic flow encounters a sharp convex corner or a smooth expansion, causing the flow to expand and accelerate
• Expansion waves are isentropic and result in a decrease in pressure, density, and temperature, while increasing the Mach number

Prandtl-Meyer function definition

• The Prandtl-Meyer function ($\nu$) is a measure of the flow deflection angle in an isentropic expansion
• It is defined as $\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}}\tan^{-1}\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)}-\tan^{-1}\sqrt{M^2-1}$
• The Prandtl-Meyer function relates the Mach number to the flow deflection angle in an isentropic expansion

Mach number vs Prandtl-Meyer function

• The Prandtl-Meyer function is a monotonically increasing function of the Mach number
• As the Mach number increases, the Prandtl-Meyer function also increases, indicating a larger flow deflection angle
• The inverse of the Prandtl-Meyer function can be used to determine the Mach number for a given flow deflection angle

Expansion wave geometry

• Expansion waves are centered at the corner or the start of the expansion and propagate into the flow at the Mach angle ($\mu$)
• The Mach angle is related to the Mach number by $\mu = \sin^{-1}(\frac{1}{M})$
• As the Mach number increases, the Mach angle decreases, and the expansion waves become more closely spaced

Expansion wave equations

• The expansion wave equations relate the upstream and downstream flow properties in an isentropic expansion
• The equations are derived from the conservation of mass, momentum, and energy, along with the isentropic flow relations
• The expansion wave equations can be used to calculate the downstream Mach number, pressure, density, and temperature for a given upstream Mach number and flow deflection angle

Mach number relations across expansion waves

• The upstream and downstream Mach numbers across an expansion wave are related by the Prandtl-Meyer function
• The downstream Mach number is always greater than the upstream Mach number, as the flow accelerates through the expansion
• The change in Mach number across an expansion wave depends on the flow deflection angle and the specific heat ratio of the gas

Pressure ratio across expansion waves

• The pressure ratio across an expansion wave is a function of the upstream Mach number and the flow deflection angle
• The pressure ratio decreases as the flow deflection angle increases, indicating a stronger expansion
• The pressure ratio across an expansion wave is always less than 1, as the pressure decreases across the expansion

Density ratio across expansion waves

• The density ratio across an expansion wave is a function of the upstream Mach number and the flow deflection angle
• The density ratio decreases as the flow deflection angle increases, indicating a stronger expansion
• The density ratio across an expansion wave is always less than 1, as the density decreases across the expansion

Supersonic nozzle flow

• Supersonic nozzle flow occurs in converging-diverging nozzles, which are used to accelerate a flow from subsonic to supersonic speeds
• The flow in a supersonic nozzle is governed by the principles of isentropic flow, choking, and shock wave formation

Converging-diverging nozzle geometry

• A converging-diverging nozzle consists of a converging section, where the flow accelerates from subsonic to sonic speeds, and a diverging section, where the flow further accelerates to supersonic speeds
• The throat is the location of minimum cross-sectional area, where the Mach number is equal to 1 (sonic conditions)
• The area ratio between the exit and the throat determines the exit Mach number for isentropic flow

Isentropic flow in nozzles

• Isentropic flow assumes that the flow is adiabatic (no heat transfer) and reversible (no entropy change)
• In an ideal converging-diverging nozzle, the flow is isentropic throughout the nozzle
• The isentropic flow relations can be used to calculate the pressure, density, and temperature ratios as functions of the Mach number

Choked flow conditions

• Choked flow occurs when the Mach number at the throat of a converging-diverging nozzle reaches 1 (sonic conditions)
• Once the flow is choked, the mass flow rate through the nozzle is at its maximum value and becomes independent of the downstream pressure
• The critical pressure ratio required for choking is given by $\frac{p_t}{p_0} = \left(\frac{\gamma+1}{2}\right)^{\frac{\gamma}{\gamma-1}}$, where $p_t$ is the throat pressure and $p_0$ is the stagnation pressure

Nozzle flow regimes

• Subsonic flow: The flow is subsonic throughout the nozzle, and the exit pressure is equal to the back pressure
• Isentropic supersonic flow: The flow is subsonic in the converging section, sonic at the throat, and supersonic in the diverging section, with the exit pressure equal to the back pressure
• Over-expanded flow: The exit pressure is greater than the back pressure, causing shock waves to form in the diverging section
• Under-expanded flow: The exit pressure is less than the back pressure, causing expansion waves to form at the nozzle exit

Over-expanded vs under-expanded nozzles

• An over-expanded nozzle has an exit pressure greater than the back pressure, resulting in shock waves in the diverging section and a decrease in exit velocity
• An under-expanded nozzle has an exit pressure less than the back pressure, resulting in expansion waves at the nozzle exit and an increase in exit velocity
• Optimal expansion occurs when the exit pressure is equal to the back pressure, resulting in isentropic supersonic flow throughout the nozzle

Shock waves in nozzles

• Shock waves can form in the diverging section of a supersonic nozzle when the exit pressure is greater than the back pressure (over-expanded flow)
• Normal shock waves cause a sudden decrease in Mach number and an increase in pressure, density, and temperature
• Oblique shock waves can form in the diverging section, causing a decrease in Mach number and a change in flow direction
• Shock waves in nozzles lead to losses in thrust and efficiency and should be avoided by proper nozzle design and operation

Supersonic airfoil theory

• Supersonic airfoil theory describes the aerodynamic characteristics of airfoils in supersonic flow
• The key aspects of supersonic airfoil theory include thin airfoil theory,